Answered

A woodworking artist makes two types of carvings: type X and type Y. He spends 3 hours making each type [tex]$X$[/tex] carving and 2 hours making each type [tex]$Y$[/tex] carving, and he can spend up to 36 hours each week making carvings. His materials cost him [tex]$\$[/tex]4[tex]$ for each type X carving and $[/tex]\[tex]$5$[/tex] for each type Y carving, and he must keep his weekly cost for materials to [tex]$\$[/tex]100[tex]$ or less. If $[/tex]x[tex]$ is the number of type X carvings he makes in a week and $[/tex]y[tex]$ is the number of type $[/tex]Y[tex]$ carvings he makes in a week, which of the following systems of inequalities models this situation?

A. $[/tex]3x + 2y \geq 36, 4x + 5y \geq 100[tex]$

B. $[/tex]3x + 2y \geq 36, 4x + 5y \leq 100[tex]$

C. $[/tex]3x + 2y \leq 36, 4x + 5y \leq 100[tex]$

D. $[/tex]3x + 2y \leq 36, 4x + 5y \geq 100$



Answer :

GinnyAnswer
To solve the problem of modeling the restrictions on the woodworking artist's time and material costs as a system of inequalities, we need to carefully translate the given constraints into mathematical inequalities.

Let's break down the problem step-by-step:

### 1. Time Constraint

The artist spends:
- 3 hours for each type [tex]\(X\)[/tex] carving.
- 2 hours for each type [tex]\(Y\)[/tex] carving.

The total time he can spend in a week is limited to 36 hours. Therefore, the inequality representing this constraint is:
[tex]\[ 3x + 2y \leq 36 \][/tex]

### 2. Cost Constraint

The cost of materials is:
- \[tex]$4 for each type \(X\) carving. - \$[/tex]5 for each type [tex]\(Y\)[/tex] carving.

The total cost for materials each week must not exceed \$100. Therefore, the inequality representing this constraint is:
[tex]\[ 4x + 5y \leq 100 \][/tex]

Putting these two constraints together, we have the system of inequalities:
[tex]\[ \begin{cases} 3x + 2y \leq 36 \\ 4x + 5y \leq 100 \end{cases} \][/tex]

### Conclusion

Based on the provided options, the correct system of inequalities that models this situation is:
[tex]\[ \begin{cases} 3x + 2y \leq 36 \\ 4x + 5y \leq 100 \end{cases} \][/tex]

Therefore, the correct answer is:
C. [tex]\( 3x + 2y \leq 36, 4x + 5y \leq 100 \)[/tex]

Other Questions